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Technical Briefs

# Minimum Move-Vibration Control for Flexible Structures

[+] Author and Article Information
Abhishek Dhanda1

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305adhanda@stanfordalumni.org

Gene Franklin

Department of Electrical Engineering, Stanford University, Stanford, CA 94305franklin@ee.stanford.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 130(3), 034503 (May 01, 2008) (6 pages) doi:10.1115/1.2807149 History: Received November 09, 2005; Revised April 18, 2007; Published May 01, 2008

## Abstract

In the rest-to-rest motion of a flexible structure with limited torque or force, it is sometimes important to control the vibrations during the move trajectory as well as at the final time. In order to achieve this control, the duration of the move must be set to be longer than the theoretical minimum time. In this paper, three definitions of trajectory vibration performance are given and a new method for input design is described, which is based on switched control. An analytical expression is obtained for the solution for lightly damped fourth-order systems, which enables a very efficient control implementation of approximately time-optimal moves. The switched control approach is generalized for presenting simplified approximate solutions for the tracking problem and other optimal control problems with singular arcs.

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## Figures

Figure 1

Impulse and step responses: (a) Posicast, (b) partial sum (PS), (c) specified negative amplitude (SNA), and (d) unity magnitude (UM)

Figure 2

Two-mass example

Figure 3

Control input and corresponding position error: (i) optimal (Vpos=10.39), (ii) time optimal (Vpos=16.72), and (iii) MPMV (Vpos=10.83). The system parameters are N=5, ω1=1, ζ=0, yref=20, and tf=10.

Figure 4

Comparison of various control schemes based on input filters for velocity saturation: (a) control input, (b) unity-magnitude (UM) input, (c) partial-sum (PS) input, “overcurrenting” (2) shown by shaded area, and (d) preloading control

Figure 5

Position move vibration versus settling time for different unit step inputs derived for the 0→1 transition. The vibration for the constant acceleration portion is calculated with Posicast settling time as the reference, i.e., Vnet=Vpos+(tPosicast−tf).

Figure 6

Residual vibration sensitivity for MPMVS and other shapers designed for ω1=1 and ζ=0. tf=2.5 for MMVS and SNA. RMPMVS is the robust counterpart of MPMVS obtained by imposing zero derivative constraints.

Figure 7

Comparison of UM-MPMVS and preload time-optimal control, and position error. UM-MPMVS is designed for 2% time relaxation as compared to preload time-optimal control. ω1=1.02, ω2=4.04, and yref=20.

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